Optimal. Leaf size=102 \[ \frac{5 b^{7/4} \sqrt{1-\frac{b x^4}{a}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{21 a^{7/4} \sqrt{a-b x^4}}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}-\frac{\sqrt{a-b x^4}}{7 a x^7} \]
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Rubi [A] time = 0.033461, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {325, 224, 221} \[ \frac{5 b^{7/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt{a-b x^4}}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}-\frac{\sqrt{a-b x^4}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 325
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{x^8 \sqrt{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{7 a x^7}+\frac{(5 b) \int \frac{1}{x^4 \sqrt{a-b x^4}} \, dx}{7 a}\\ &=-\frac{\sqrt{a-b x^4}}{7 a x^7}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}+\frac{\left (5 b^2\right ) \int \frac{1}{\sqrt{a-b x^4}} \, dx}{21 a^2}\\ &=-\frac{\sqrt{a-b x^4}}{7 a x^7}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}+\frac{\left (5 b^2 \sqrt{1-\frac{b x^4}{a}}\right ) \int \frac{1}{\sqrt{1-\frac{b x^4}{a}}} \, dx}{21 a^2 \sqrt{a-b x^4}}\\ &=-\frac{\sqrt{a-b x^4}}{7 a x^7}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}+\frac{5 b^{7/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt{a-b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0101221, size = 52, normalized size = 0.51 \[ -\frac{\sqrt{1-\frac{b x^4}{a}} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};\frac{b x^4}{a}\right )}{7 x^7 \sqrt{a-b x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 109, normalized size = 1.1 \begin{align*} -{\frac{1}{7\,a{x}^{7}}\sqrt{-b{x}^{4}+a}}-{\frac{5\,b}{21\,{x}^{3}{a}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{5\,{b}^{2}}{21\,{a}^{2}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b x^{4} + a}}{b x^{12} - a x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.75743, size = 46, normalized size = 0.45 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x^{7} \Gamma \left (- \frac{3}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-b x^{4} + a} x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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